218 11 Change of Num ́eraire
arbitrage stands for the existence of an equivalent risk neutral (i.e. for a local
martingale) measure.
It turns out that the problem is related to the characterisation of those
contingent claims that can be hedged. This topic was studied by Jacka [J 92]
and Ansel-Stricker [AS 94]. These authors use the H^1 -BMO duality. We will
give a measure independent characterisation in terms of maximal elements of
attainable claims. These elements were already used, as a technical device,
in Chap. 9. The proofs of the theorems below use these results as well as an
extension of a duality relation from Delbaen [D 92].
The technique of a change of num ́eraire together with the change of the
risk neutral measure was used by El Karoui, Geman and Rochet [EGR 95] and
Jamshidian [J 87] to facilitate calculations of prices of contingent claims*.
The results of this paper can also be used to build consistent models of
exchange rates of currencies. In this case the discounting procedure depends
on the currency since the interest rate in different currencies will be different.
We refer to Delbaen-Shirakawa [DSh 96] for details.
The rest of this section is devoted to the introduction of the basic nota-
tion. Sect. 11.2 recalls known facts from arbitrage theory. In Sect. 11.3 we
extend the duality equality and relate it to properties of maximal elements.
Sect. 11.4 finally contains the main theorem on the change of num ́eraire and
the application to the theory of hedgeable elements.
The setup in this paper is the usual setup in mathematical finance. A prob-
ability space (Ω,F,P) with a filtration (Ft) 0 ≤tis given. In order to cover the
most general case, the time set is supposed to beR+. The filtration is assumed
to satisfy the “usual conditions”, i.e. it is right continuous andF 0 contains all
null sets ofF. A price processS, describing the evolution of the discounted
price ofdassets, is defined onR+×Ω and takes values inRd.Inorderto
use the results of Chap. 9, we suppose that the processSis locally bounded.
This assumption is fairly general, in particular it covers the case of continuous
price processes. As shown under a wide range of hypotheses, the assumption
thatSis a semi-martingale follows from arbitrage considerations. We can
therefore assume that the processSis a semi-martingale. Since it is also lo-
cally bounded it is a special semi-martingale. Stochastic integration is used to
describe outcomes of investment strategies. When dealing with processes in
dimension higher than 1 it is understood that vector stochastic integration is
used. We refer to Protter [P 90] and Jacod [J 79] for details on these matters.
The authors want to thank Ch. Stricker and H. Shirakawa for helpful
discussions on the topic. Part of the research was done while the first author
was on visit in the University of Tokyo. Discussions with the colleagues and
especially with S. Kusuoka, S. Kotani and N. Kunitomo contributed to the
development of this paper.
∗Note added in this reprint: The idea of changing the num ́eraire can be traced
back to the work of Margrabe [M 78a], [M 78b]